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The first-order linear differential equation y' + P(x)y = Q(x) may be transformed into a separable differential equation by multiplying both sides by the integrating factor e^integral P(x)dx.
A first-order linear differential equation is an equation of the form:
y' + P(x)y = Q(x)
where P and Q are continuous functions.
A Polar equation that has one of the four forms:
r = de/(1+/-ecos 0)
r = de/(1+/-esin 0)
is a conic section. The conic is a parabola if e = 1, an ellipse if 0<e<1, or a hyperbola if e>1
Let F be a fixed point and l a fixed line in a plane. The set of all points P in the plane, such that the ratio d(P,F)/d(P,Q) is a positive constant e with d(P,Q) the distance from P to l, is a conic section. The conic is a parabola if e = 1, an ellipse if 0<e<1, and a hyperbola if e>1
ds = Root(r^2 + (dr/d0)^2)d0
If f is continuous and f(0)./= 0 on [alpha,beta], where 0</=alpha</=beta</=2pi, then the are A of the region bounded by the graphs of r = f(0), 0 = alpha, and 0 = beta is:
A = Integral from alpha to beta of 1/2[f(0)]^2d0 =
Integral from alpha to beta of 1/2 r^2d0
The slope m of the tangent line to the graph of r = f(0) at the point P(r,0) is:
m = ((dr/d0)sin 0 +r cos 0)/((dr/d0)cos 0 - r sin 0)
i. The graph of r = f(0) is symmetric with respect to the polar axis if substitution of -0 for 0 leads to an equivalent equation.
ii. The graph of r = f(0) is symmetric with respect to the vertical line 0 = pi/2 if substitution of either (a)pi - 0 for 0 or (b) -r for r and -0 for 0 leads to an equivalent equation.
iii. The graph of r = f(0) is symmetric with respect to the pole if substitution of either (a) -r for r or (b) pi + 0 for 0 leads to an equivalent equation.
The rectangular coordinates (x,y) and polar coordinates (r,0) of a point P are related as follows:
i. x = r cos 0, y = r sin 0
ii. r^2 = x^2 + y^2, tan 0 = y/x if x doesn't = 0
If a smoot curve C is given parametrically by x = (ft), y = g(t); a</=t</=b, and if C does not intersect itself, except possibly for t = a and t = b, then the length L of C is:
L = Integral from a to b of Root([f'(t)]^2 + [g'(t)]^2dt =
Integral from a to b of Root((dx/dt)^2 + (dy/dt)^2)dt
d^2y/dx^2 = d/dx(y') = (dy'/dt)/(dx/dt)
If a smooth curve C is given parametrically by x = f(t), y = g(t), then the slope dy/dx of the tangent line to C at P(x,y) is:
dy/dx = (dy/dt)/(dx/dt), provided dx/dt doesn't = 0
Let C be the curve consisting of all ordered pairs (f(t), g(t)), where f an g are continuous on an interval I. The equations:
x =f(t), y = g(t)
for t in I, are parametric equations for C with parameter t.
A Plane curve is a set C of ordered pairs (f(t),g(t)) where f and g are continuous functions on an interval I.
The graph of the equation:
Ax^2 + Bxy + Cy^2 + Dx +Ey + F = 0
is either a conic or a degenerate conic. If the graph is a conic, then it is:
i. a parabola if B^2 -4AC = 0
ii. an ellipse if B^2-4AC < 0
iii. a hyperbola if B^2-4AC > 0
To eliminate the xy-term from the equation:
Ax^2 + Bxy + Cy^2 + Dx +Ey + F = 0,
where B doesn't = 0, choose an angle Q such that
cot 2Q = (A-C)/B with 0^o<2Q<180 degrees
and use the rotation of axes formulas.
If the x- and y-axes are rotated about the origin O, through an acute angle Q, then the coordinates (x,y) and (x',y') of a point P in the xy- and x'y'-planes are related as follows:
i. x = x'cos Q - y'sin Q, y = x'sinQ + y'cos Q
ii. x' = xcos Q + ysin Q, y' = -xsin Q + ycos Q
The graph of the equation:
y^2/a^2 - x^2/b^2 = 1
is a hyperbola with vertices (0,+/-a). The foci are (0,+/-c), where c^2 = a^2 + b^2
The graph of the equation:
x^2/a^2 - y^2/b^2 = 1
is a hyperbola with vertices (+/-a,0). The foci are (+/-c,0) where c^2 = a^2 + b^2
A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane (the foci) is a positive constant.
The eccentricity e of an ellipse is:
e = c/a = Root(a^2-b^2)/a
The graph of the equation:
x^2/b^2 + y^2/a^2 = 1
for a^2> b^2 is an ellipse with vertices (0,=/-a). The endpoints of the minor axis are (+/-b,0). The foci are (0,+/-c) where c^2 = a^2 - b^2
The graph of the equation:
x^2/a^2 + y^2/b^2 = 1
for a^2>b^2 is an ellipse with vertices (+/-a,0). The endpoints of the minor axis are (0,=/-b). The foci are (=/-c,0) where c^2 = a^2 - b^2.
An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (the foci) in the plane is constant.
If (x,y) are the corrdinates of a point P in an xy-plane and if (x',y') are the coordinates of P in an x'y'-plane with origin at the point (h,k) of the xy-plane, then:
i. x = x' + h, y = y' + k
ii. x' = x - h, y' = y - k
A parabola is the set of all points in a plane equidistant from a fixed point F(the focus) and a fixed line l (the directrix) in the plane.
If lxl < 1, then for every real number k,
(1+x)^k = 1 + kx +K(k-1)/2!(x^2) + ....+(k(k-1)....(k-n+1))/n!(x^n)+....
If x is any real number,
limit as n->infinity of lxl^n/n! = 0
Let f have derivatives of all orders throughout an interval containing c, and let Rn(x) be the Taylor remainder of f at c. If:
Limit as n-> infinity of Rn(x) = 0
for every x in the interval, then f(x) is represented by the Taylor series for f(x) at c.
Let c be a real number and let f be a function that has n derivatives at c: f'(c), f''(c),....f^(n)(c). The nth-degree Taylor polynomial Pn(x) of f at c is:
Pn(x) = f(c) + f'(c)(x-c) + f''(c)/2!(x-c)^2 + .... + f^(n)(c)(x-c)^n /n!
If a function f has a power series representation:
f(x) = (sigma)An x^n
with radius of convergence r>0, then f^(k) (0) exists for every positive integer k and An = f^(n) (0)/n!. Thus:
f(x) = f(0) + f'(0)x + f''(0)/2! (x^2) +.....+ f^(n) (0)/n! (x^n) +....
If x is any real number,
e^x = 1 + x + x^2/2! + x^3/3! + ...+ x^n/n! + ....
If (sigma)Anx^n is a power series, then exactly one of the following is true:
i. The series converges only if x = 0
ii. The series is absolutely convergent for every x
iii. There is a number r > 0 such that the series is abs. convergent if x is in the open interval (-r,r) and divergent if x < -r or x > r
i. If a power series (sigma)Anx^n converges for a nonzero number c, then it is absolutely convergent whenever lxl < lcl
ii. If a power series (sigma)Anx^n diverges for a nonzero number d, then it diverges whenever lxl > ldl
Let x be a variable. A power series in x is a series of the form (sigma)An x^n = A0 + A1x + A2x^2 + ...+ Anx^n
where each Ak is a real number.
If a series (sigma)An is absolutely convergent, then (sigma)An is convergent.
A series (sigma)An is conditionally convergent if (sigma)An is convergent and (sigma)lAnl is divergent
A series (sigma)An is absolutely convergent if the series (Sigma)lAnl = lA1l + lA2l + ... + lAnl
is convergent.
The alternating series:
(sigma)(-1)^n-1(An) = A1 - A2 + A3 - A4 + ....+ (-1)^n-1(An)
is convergent if the following 2 conditions are satisfied:
i. Ak > Ak+1 > 0 for every k
ii. Limit of An = 0
Let (sigma)An be a pisitive-term series, and suppose
Limit as n-> infinity of ^nRoot(An) = L
i. If L < 1, the series is convergent
ii. If L > 1, the series is divergent
iii. If L = 1, apply a different test, the series can be either convergent or divergent.
Let (sigma)An be a pisitive-term series, and suppose
Limit as n-> infinity of An +1/An = L
i. If L < 1, the series is convergent
ii. If L > 1, the series is divergent
iii. If L = 1, apply a different test, the series can be either convergent or divergent.
Let (sigma)An and (sigma)Bn be positive-term series. If:
Limit as n->infinity of An/Bn = c>0,
then either both series converge or diverge.
Let (sigma)An and (sigma)Bn be positive-term series:
i. If (sigma)Bn converges and An</=Bn for every positive integer n, then (sigma)An converges
ii. If (sigma)Bn diverges and An>/= Bn for every positive integer n, then (sigma)An diverges.
The p-series (sigma)1/n^p
i. converges if p>1
ii. diverges if p</= 1
A p-series, or a hyperharmonic series, is a series of the form:
(sigma)1/n^p = 1 + 1/2^p + 1/3^p + ... + 1/n^p
where p is a positive real number.
If (sigma)An is a series, let f(n) = An and let f be the function obtained by replacing n with x. If f is positive-valued, continuous, and decreasing for every real number x>/= 1, then the sereis (sigma)An
i. converges if Integral from 1 to infinity of f(x)dx converges
ii. diverges if integral from 1 to infinity of f(x)dx diverges
If (sigma)An is a positive-term series and if there exists a number M such that:
Sn = a1 +a2 + ... + An <M
for every n, then the series converges and has a sum S</= M. If no such M exists, the series diverges.
If (sigma)An is a convergent series and (sigma)Bn is divergent, then (sigma)(An + Bn) is divergent.
If (sigma)An and (Sigma)Bn are convergent series with sums A and B respectively, then
i. (sigma)(An +Bn) converges and has sum A + B
ii. (sigma)cAn converges and has sum cA for every real number c
iii. (sigma)(An-Bn) converges and has sum A - B
If (sigma)An and (sigma)Bn are series such that Aj = Bj for every j> k, where k is a positive integer, then both series converge or both series diverge.
If a series (sigma)An is convergent then the limit An = 0
Let a not = 0. The geometric series
a + ar + ar^2 + ... + ar^n-1
i. converges and has the sum S = a/(1-r) if lrl < 1
ii. diverges if lrl >/= 1
The harmonic series is the divergent series
1 + 1/2 + 1/3 + ... + 1/n
A series (sigma)An is convergent if its sequence of partial sums {Sn} converges - that is, if:
limit as n->infinity of Sn = S for some real number S
The limit S is the sum of the series (sigma)An and we write
S = a1 + a2 + ... + an + ...
The series (sigma)An is divergent if {Sn} diverges. A divergent series has no sum.
i. The kth partial sum Sk of the series (sigma)An is:
Sk = a1 + a2 + .... + ak
ii. the sequence of partial sums of the series (sigma)An is
S1, S2, S3, ..... , Sn,....
An infinite series is an expression of the form:
a1+a2+a3+.....+an+....
or, in summation notation,
(Sigma)An
Each number Ak is a term of the series, and An is the nth term.
If a nonempty set S of real numbers has an upper bound, then S has a least upper bound.
A bounded, monotonic sequence has a limit.
Let {An} be a sequence. If limit as n->Infinity of lAnl = 0, then limit as n->Infinity of An=0
If {An}, {Bn}, and {Cn} are sequences and An</= Bn</=Cn for every n and if:
limit as n->Infinity of An = L = limit as n->Infinity of Cn,
then:
limit as n->Infinity of Bn = L
i. limit as n->infinity of r^n = 0 if lrl<1
ii. limit as n->infinity of lr^nl = infinity if lrl>1
Let {An} be a sequence, let f(n) = An, and suppose that f(x) exists for every real number x>/= 1.
i. If limit as x-> Infinity of f(x) = L, then limit as n-> Infinity of f(n) = L
ii. If limit as x-> infinity of f(x) = infinity or - infinity, then limit as n-> Infinity of f(n) = infinity or -infinity.
The notation:
Limit as n-> infinity of An = infinity
means that for every positive real number P there exists a number N such that An>P whenever n>N
A sequence {An} has the limit L, or converges to L, denoted by either:
Limit as n-> infinity of An = L or An-> L as n -> Infinity,
if for every E> 0 there exists a positive number N such that:
lAn - Ll<E whenever n>N.
If such a number L does not exist, the sequence has no limit, or diverges.
A sequence is a function f whose domain is the set of positive integers.
Sequence notation:
a1,a2,a3,a4,.......,an,.......
Let y be a differntiable function of t such that y>0 for every t, and let y_0 be the value of y at t = 0. If dy/dt = cy for some constant c, then:
y = y_0e^ct
i. lim as h-> 0 (1+h)^1/h = e
ii. lim as n->infinity of (1+1/n)^n = e
i.Dx log_a x = Dx (Ln x/Ln a) = 1/Ln a (1/x)
ii. Dx log_a lul = Dx (Ln lul/Ln a) = 1/Ln a (1/u)Dxu
y = log_a x if and only if x =a^y
i. integral of a^x dx = (1/Ln a) a^x + C
ii. integral of a^u du = (1/Ln a) a^u + C
i. Dx a^x = a^x Ln a
ii. Dx a^u = (a^u Ln a)Dx u
Let a>0 and b>0. If u and v are any real numbers, then:
a^u(a^v) = a^u+v
(a^u)^v = a^uv
(ab)^u = a^ub^u
a^u / a^v = a^u-v
(a/b)^u = a^u / b^u
a^x = e^xlna
for every a>0 and every real number x
i. integral of tan u du = -Ln lcos ul + C
ii. integral of cot u du = Ln lsin ul + C
iii. integral of sec u du = Ln lsec u + tan ul + C
iv. integral of csc u du = Ln lcsc u - cot ul + C
If u = g(x) and g is differentiable, then
integral of e^u = e^u + C
If u = g(x) and doesn't = 0 and g is differentiable, then:
intregal of 1/u du = ln lul + C
If u = g(x) and g is differentiable, then:
Dx e^u = e^u Dx u
Dx e^x = e^x
If p and q are real numbers and r is a rational number, then:
i. e^p(e^q) = e^p+q
ii. e^p / e^q = e^p-q
iii. (e^p)^r = e^pr
i. ln e^x = x for every x
ii. e^ln x = x for every x > 0
Definition of e^x: If x is any real number, then
e^x = y if and only if ln y = x
e=2.71828
The letter e denotes the positive real number such that ln e = 1.
The natural exponential function, denoted by exp, is the inverse of the natural logarithmic function.
To every real number x there corresponds exactly one positive real number y such that ln y = x.
If u=g(x) and g is differentiable, then
I) Dx ln u = 1/u Dx u if g(x)>0
II) Dx ln |u| = 1/u Dx u if g(x) cannot equal 0
Dx ln x = 1/x
The natural logarithmic function, denoted by ln, is defined by
ln x= integral x to 1 1/t dt
for every x>0
domain of f^-1 = range of f
range of f^-1 = domain of f
Theorem 7.3 Let f be a one-to-one function with domain D and range R. if g is a function with domain R and range D, then gi is the inverse function of f if and only if both the following conditions are true: I. g(f(x))=x for every x in D II. f(g(y))=y for every y in R
Definition 7.2 Let f be one-to-one function with domain D and range R. A function g with domain R and range D is the inverse funtion of f, provided the following condition is tru for every x in D and every y in R: y=f(x) if and only if x=g(y)
Definition 7.1 A function f with domain D and range R is a one-to-one funtion if whenever (a) does not equal (b) in D, then f(a) cannot equal f(b) in R.
i. Integral of 1/Root(a^2+u^2))du = sinh^-1 u/a + C, a>0
ii. Integral of 1/Root(u^2-a^2)du = cosh^-1 u/a + C, 0<a<u
iii. Integral of 1/(a^2 - u^2)du = 1/a tanh^-1 u/a + C, lul < a
iv. Integral of 1/(uRoot(a^2 - u^2))du = -1/a sech^-1 lul/a + C, 0<lul<a
i. Dx sinh^-1 u = 1/Root(u^2 + 1))Dx u
ii. Dx cosh^-1 u = 1/Root(u^2 - 1))Dx u, u>1
iii. Dx tanh^-1 u = 1/(1-u^2)Dx u , lul <1
iv. Dx sech^-1 u = -1/(uRoot(1-u^2))Dx u, 0<u<1
i. sinh^-1 x = Ln(x + Root(x^2 + 1))
ii. cosh^-1 x = Ln(x + Root(x^2 - 1)), x>x/= 1
iii. tanh^-1 x = 1/2 Ln (1+x)/(1-x), lxl < 1
iv. sech^-1 x = Ln (1 + Root(1 - x^2))/x, 0<x</= 1
i. Integral of sinh u du = cosh u + C
ii. Integral of cosh u du = sinh u + C
iii. Integral of sech^2 u du = tanh u + C
iv. Integral of csch^2 u du = -coth u + C
v. Integral of sech u tanh u du = -sech u + C
vi. Integral of csch u coth u du = -csch u + C
i. Dx sinh u = cosh u Dx u
ii.Dx cosh u = sinh u Dx u
iii. Dx tanh u = sech^2 u Dx u
iv. Dx coth u = -csch^2 u Dx u
v. Dx sech u = -sech u tanh u Dx u
vi. Dx csch u = -csch u coth u Dx u
i. 1 - tanh^2 x = sech^2 x
ii. coth^2 x - 1 = csch^2 x
i. tanh x = sinh x/cosh x = (e^x - e^-x)/(e^x + e^-x)
ii. coth x = cosh x/sinh x = (e^x + e^-x)/(e^x - e^-x) , x doesn't = 0
iii. sech x = 1/cosh x = 2/(e^x + e^-x)
iv. csch x = 1/sinh x = 2/(e^x - e^-x) , x doesn't = 0
cosh^2 x - sinh^2 x = 1
The hyperbolic sine function, denoted by sinh, and the hyperbolic cosine function, denoted by cosh are defined by:
sinh x = (e^x -e^-x)/2 and cosh x = (e^x + e^-x)/2
for every real number x
i. Integral of 1/(Root(a^2-u^2))du = sin^-1 u/a + C
ii. Integral of 1/(a^2+u^2)du = 1/a (tan^-1 (u/a)) + C
iii. Integral of 1/u(Root(u^2-a^2))du = 1/a (sec^-1 (u/a)) + C
i. Dx sin^-1 u = 1/(Root(1-u^2))Dx u
ii. Dx cos^-1 u = -1/(Root(1-u^2))Dx u
iii. Dx tan^-1 u = 1/(1+u^2)Dx u
iv. Dx sec^-1 u = 1/u(Root(U^2 - 1))Dx u
The inverse secant function, or arcsecant function, denoted by sec^-1, or arcsec, is defined by:
y = sec^-1 x = arcsec x if and only if x = sec y
for lxl>/= 1 and y in [0,pi/2) or in [pi,3pi/2)
i. tan(tan^-1 x) =tan(arctan x) = x for every x
ii. tan^-1(tan x) = arctan(tan x) = x if -pi/2<x<pi/2
The inverse tangent function, or arctangent function, denoted by tan^-1, or arctan, is defined by:
y = tan^-1 x =arctan x if and only if x = tan y
for every x and -pi/2<y<pi/2
i. cos(cos^-1 x) = cos(arccos x) = x if -1</=x</=1
ii. cos^-1(cos x) = arccos(cos x) = x if 0</=x</=pi
The inverse cosine function, denoted cos^-1, is defined by:
y = cos^-1 x if and only if x = cos y
for -1</=x</=1 and 0</=y</=pi
i.sin(sin^-1 x) = sin(arcsin x) = x if -1</=x</=1
ii. sin^-1(sin x) = arcsin(sin x ) = x if -pi/2</=x</=pi/2
The inverse sine function, denoted sin^-1, is defined by:
y = sin^-1 x if and only if x = sin y
for -1</=x</=1 and -pi/2</=y</=pi/2
If an integrand is a rational expression in sin x and cos x, the following substitutions will produce a rational expression in u:
sin x = 2u/(1+u^2)
cos x = (1-u^2)/(1+u^2)
dx = 2/(1+u^2)du
where u = tan x/2
If u = f(x) and v = g(x) and if f' and g' are continuous, then:
Integral of u dv = uv - Integral of v du
If f has a discontinuity at a number c in the open interval (a,b) but is continuous elsewhere on [a,b], then:
Integral from a to b of f(x)dx = Integral from a to c of f(x)dx + Integral from c to b of f(x)dx,
provided both of the improper integrals on the right converge. If both converge, then the value of the improper integral from a to b of f(x)dx is the sum of the 2 values.
i. If f is continuous on [a,b) and discontinuous at b, then:
Integral from a to b of f(x)dx = limit as t->b^- of integral from a to t of f(x)dx,
provided the limit exists.
ii. If f is continuous on (a,b] and discontinuous at a, then:
Integral from a to b of f(x)dx = limit as t->a^+ of integral from t to b of f(x)dx,
provided the limit exists.
Let f be continuous for every x. If a is any real number, then:
Integral from -infinity to infinity of f(x)dx = integral from -infinity to a of f(x)dx + Integral from a to infinity of f(x)dx.
provided both of the improper integrals on the right converge.
i. If f is continuous on [a,Infinity), then:
Integral from a to infinity of f(x)dx = limit as t->infinity of the integral of f(x)dx,
provided the limit exists.
ii. If f is continuous on (-infinity,a], then:
Integral from - infinity to a of the integral of f(x)dx = limit as t->-infinity of the integral of f(x)dx,
provided the limit exists.
Suppose f and g are differentiable on an open interval (a,b) containing c, except possibly at c itself. If f(x)/g(x) has the indeterminate form 0/0 or Infinity/Infinity at x=c and if g'(x) doesnt = 0 for x doesn't = c, then:
limit as x->c f(x)/g(x) = limit x->c f'(x)/g'(x)
provided either limith as x->c f'(x)/g'(x) exists or limit as x->c f'(x)/g'(x) = Infinity
If f and g are continuous on [a,b] and differentiable on (a,b) and if g'(x) doesn't = 0 for every x in (a,b), then there is a number w in (a,b) s.t.:
(f(b)-f(a))/(g(b)-g(a)) =f'(w)/g'(w)
